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The residence time of solids in gas-fluidized beds
Abstract
A critical survey is given of the different models proposed for solids transport in gas-fluidized beds. It is shown that many of the models can fit the residence time distribution curve—the Ft-t curve—fairly well despite the fact that the physical behaviour of the bed is not recognized in these models. The intensity curve is recommended as a good tool for comparison of the models with the experiments.
... 5 Kunii and Levenspiel 9 showed that, at gas velocities close to u mf , the segregation of jetsam particles is more dominant than at higher velocities. Verloop et al. 10 reported that gas fluidized beds approach perfect mixing for u 0 > 1.5u mf , whereas other investigators gave much higher values. Peeler and Huang, 4 for example, investigated the mixing of sand particles having particle sizes of 208 and 2020 µm and found perfect mixing for u 0 > 15u mf . ...
... To investigate the reproducibility of the experiments, the measurements were rerun four times as shown in Figure 4, and it is clear that the results are very reproducible. The experimental data follow the curve calculated with which is in accordance with Verloop et al. 10 and Levenspiel, 24 giving the equation for a continuously stirred tank reactor (CSTR) in series with a plug-flow reactor (PFR). The parameter z gives the time when the first particle exits the siphon and was found to be quite constant with According to Verloop et al. 10 and Levenspiel, 24 z gives the residence time of particles in the PFR, t res,PFR , ...
... The experimental data follow the curve calculated with which is in accordance with Verloop et al. 10 and Levenspiel, 24 giving the equation for a continuously stirred tank reactor (CSTR) in series with a plug-flow reactor (PFR). The parameter z gives the time when the first particle exits the siphon and was found to be quite constant with According to Verloop et al. 10 and Levenspiel, 24 z gives the residence time of particles in the PFR, t res,PFR , ...
A process is under development for the steam gasification of biomass to produce a hydrogen-rich gas for use with a fuel cell to generate electricity on a local scale. A pilot plant is currently under construction in southern Italy operated with a circulating fluidized bed, and to predict the fluid dynamic conditions within the plant, a cold laboratory rig was built according to existing scaling laws, and experimental studies were carried out. In this paper, we present the experimental results concerning the solids residence time of particles introduced into the system and the particle mixing in the “gasifier” section of the model. Both parameters are of fundamental importance for the operation of the pilot plant as they determine the performance of the gasification process. It is shown that the biomass particles spend sufficient time in the gasifier to be fully gasified, and an equation is derived to predict the mean residence time of the biomass particles as a function of the dimensionless mass turnover of the circulating bed material. In addition, it is shown that the biomass particles are well mixed within the circulating bed material. One reason for this is a result of the geometric design of the apparatus.
... Many of the general modeling approaches developed for chemical reactors have been adapted for modeling particle RTDs in bubbling and circulating beds. Some relevant articles in the literature that discuss RTD modeling in this context include the following: Yagi and Kunii (1961a), Verloop et al (1968), Berruti et al (1988), Ambler et al (1990), Smolders and Baeyens (2000), Harris et al (2002), Bhusarapu et al (2004), and Andreux et al (2008). As with the modeling approaches used for the more general problem in chemical reactors, particle RTD models for bubbling and fluidized beds have adopted one of three basic approaches, listed below in increasing order of complexity: ...
... Also, in spite of their ideal assumptions, such models are still based on physical principles and thus can presumably provide some level of physical insight. A relatively complete summary of all the simplified RTD models that have been previously considered is given in Verloop et al (1968). The RTD approximations that appear most promising for the present modeling needs for biomass particle pyrolysis are: ...
A mathematical model for the ethylene-propylene copolymerization with a Ziegler - Natta catalyst in a gas phase fluidized bed reactor is presented. The model includes a two active site kinetic model with spontaneous transfer reactions and site deactivation. Also, it is studied and simulated the growth of a polymeric particle which is exposed to an outside atmosphere (monomers concentrations and temperature) that represent the emulsion phase conditions of the reactor. Particle growth model is the basis for the Study of the sizes distribution into the reactor. Two phase model of Kunii - Levenspiel is the basis for the modeling and simulation of the fluid bed reactor, the models developed consider two extreme cases for the gas mixed grade in the emulsion phase (perfectly mixed and plug flow). The solution of the models includes mass (for the two monomers) and energy balances, coupled with the particle growth and residence time distribution models.
Axial mixing of solids in a fluidized bed of glass beads (0.362 mm) in the slugging and turbulent flow regimes was studied in a Plexiglas column (inside diameter, 0.1 m; height, 3.0 m). The effective axial dispersion coefficient of the solid phase in the bed was determined from the axial transport of heat at steady state. The effective axial dispersion coefficient of the solid phase is almost constant in the slugging flow regime, and increases with an increase in gas velocity in the turbulent flow regime. The mixing of solids in the bed is assumed to be described by mixing tanks in series connected by perfectly mixed and plug flows. The fraction of perfectly mixed flow in the bed remains almost constant in the slugging flow regime, and increases with an increase in gas velocity in the turbulent flow regime. The effects of column size and properties of the solid particles on the effective axial dispersion coefficient of the coarse particles in the bed were determined. The effective axial dispersion dataof the solids, in terms of the Peclet number, were correlated with the Archimedes number.
The residence time distribution (RTD) of solids in spiral fluidised beds was experimentally investigated, covering as variables the flow rates of the phases, the particle characteristics and the column and plate geometries. The RTD was adequately described by the Fickian diffusion model, indicating that particle dispersion can be predicted from a knowledge of particle diffusivity. Correlations for the particle mean residence times and the particle diffusivity are presented.
An experimental study was carried out to estimate the residence time distribution of low density particles injected into a fluidized bed reactor containing sand particles. Tracer experiments were performed at room temperature, using a pilot plant pyrolysis reactor. A novel technique was used to detect and measure the flow of solid tracer particles having different physical characteristics entering and leaving the fluid bed. The experimental results demonstrated that the degree of particle entrainment was a function of the fluidizing gas velocity, the particle size and the particle density. Solid mixing, segregation and entrainment were also studied as functions of physical and operating parameters. Various models were tested to characterize the non-ideal solids flow patterns within the fluid bed. A circulation model appeared to give a good description of the physical mechanism involved and to provide the best agreement with the experimental results.
A model of the fluidized bed calcination process based on solids flow and mixing has been developed to calculate the change in the bed particle size distribution as a function of time and calciner operating conditions. The model is based on a mechanistic approach for determining the particle size distributions resulting from feed deposition, particle attrition, elutriation, seed addition and product removal. Comparison between the model results and data from a pilot plant calciner show good agreement. Because the model is based on mechanisms occurring in the fluidized bed calciner, it should be able to accurately scale up pilot plant results to predict operations in a full-scale calciner.
In a bench-scale fluidized bed reactor, 20cm in dia., residence time distribution of solid particles (dp, = 137μm) were measured by the radionuclide technique (24Na2C03) in the absence and in the presence of the chemical reaction 2NaHCO3→Na2C03+C02+H2O. The residence time distributions were evaluated by a backflow cascade model by nonlinear optimization. The radial and logitudinal concentration profile of NaHCO3 in the emulsion phase of the reactor were measured during steady-state operation. The solid is well-mixed.The connection of the measured bubble properties with the longitudinal solid dispersion coefficient and the use of the Haines-King-Woodburn model allow
Earlier work has shown that when non-segregating particles are fed to a continuous mixer the fluctuations in the output stream can be predicted if the input fluctuations and residence time distribution for the mixer are known. In this paper the work is extended to particles which segregate in the mixer. The amount of segregation occurring is characterised by a steady state variance, that is the variance of composition fluctuations in the exit stream when a mixture containing no fluctuations is fed to the mixer. It is shown that when known composition fluctuations are fed to the mixer the variance for the output stream can be predicted by adding together the steady state variance and the variance predicted for no segregation, using the residence time distribution. Predicted variances agree with the experimental results obtained for a continuous fluidized bed mixer.
The shallow fluidised bed is now a standard industrial technique for heating or cooling particulate solids. In spite of this, satisfactory design procedures for this type of equipment have not yet been formulated. Analyses are presented in this paper which can be used to design both single and double stage equipment. The analyses cover both possibilities of thermal equilibrium or non-thermal equilibrium in the bed, and also may be used for various solids flow conditions.
The properties of the tracer washout curve and the residence-time distribution and Danckwerts' F curve [3] are related. It is shown that convexity of the washout curve is associated with a strictly positive residence-time distribution (density) function. A wide variety of linear mixing phenomena possess this characteristic.
The development of a noninvasive PC‐based computer automated radioactive particle tracking (CARPT) facility for investigation of phase recirculation and turbulence in multiphase systems such as fluidized beds and bubble columns is presented. In this facility, the motion of a single radioactive particle, which is dynamically similar to the recirculating phase, is monitored by an array of scintillation detectors which surround the test section. An on‐line computer is used to map the flow field of the recirculating phase. The data acquisition is achieved by commercially available nuclear instrumentation via a modular, high‐speed GPIB‐CAMAC system through assembly language software. Using CARPT, solids’ motion in gas‐fluidized beds and liquid motion in bubble columns have been investigated. The capabilities and versatility of the CARPT facility is described by illustrating some typical results for mean recirculation in gas‐fluidized beds with and without internals and in a gas‐liquid bubble column. The results include the mean circulation profiles and turbulence parameters such as the Reynolds normal and shear stresses and the turbulent eddy dispersion coefficients. Potential applications of CARPT technique to other recirculating systems are also discussed.
An automated non-intrusive image analysis method has been developed for following the course of solids mixing in two-dimensional bubbling fluidized beds. In this investigation, experimental data have been obtained on the axial mixing of uniform solids. Oscillations in the concentration response, resulting from the gross circulation of the solids, have been observed experimentally. These oscillations become increasingly more prominent as the bed particle size increases. These measurements have been used to evaluate the three-phase counter-current back-mixing model (Gwyn et al.). The bubble parameters required for the model were obtained from independent experiments conducted as a part of this investigation; the exchange coefficient however, was found by parameter estimation using the solids mixing data. With this choice of parameters, the counter-current flow model has been found to predict the experimental trends reasonably well. The estimated values for the exchange coefficient do not compare favourably with the predictions of the models available in the literature (Yoshida and Kunii, and Chiba and Kobayashi). These models predict that the wake exchange coefficient should increase with increase in the minimum fluidization of the bed particles. Our results, on the other hand, show that the wake exchange coefficient increases with UO/Umf for UO/Umf < 3 and the values, in this region are independent of the particle size. In line with these results, the experimental measurements of Chiba and Kobayashi, for injected bubbles in a two-dimensional fluidized bed of particles smaller than those used in this investigation, are found to be in excellent agreement with the lower bound of our estimations.
In this paper we study stochastic models for the transport of particles in a fluidized bed reactor, and compute the associated residence time distribution (RTD). Our main model is basically a diffusion process in [0; A] with reflecting/absorbing boundary conditions, modified by allowing jumps to the origin as a result of transport of particles in the wake of rising fluidization bubbles. We study discrete time birth-death Markov chains as approximations to our diffusion model. For these we can compute the particle distribution inside the reactor as well as the RTD by simple and fast matrix calculations. It turns out that discretization of the reactor into a moderate number of segments already gives excellent numerical approximations to the continuous model. From the forward equation for the particle distribution in the discrete model we obtain in the diffusion limit a partial differential equation for the particle density p(t; x) @ @t p(t; x) = 1 2 @ 2 @x 2 [D(x)p(t; x)] Gamma @ @...
Such a reactor model as fulfils the eq. (2) -i.e. mean diffusivity and velocity are able to be assumed is proposed and the residence time curves of the model are mathematically solved as eq. (7)-(8)'.
Based on them, we discussed various behaviours of such a reactor model.
From comparisons between the model and the measurements performed by Gilliland et. al.4), 5), 6) mainly on the fluidized bed, we have concluded that the model is able to give fairly good representation to the experimental results.
Based on such a quantity as U, we have discussed the characteristics on the concentration driving force of various flow reactors and concluded that the term U seems to be a representative quantity of such characteristics.
A revised version of a multiloop circulation model first proposed by van de Vusse is shown to fit experimental residence time distribution data obtained for a 100 l. stirred vessel. A simple and generally applicable method for the solution of models of this type is described.RésuméOn montre une version modifiée d'un modèle de circulation à boucles multiples proposé pour la première fois par van der Vusse, pour tenir compte des données expérimentales de la distribution du temps de contact pour un récipient de 100 litres avec agitateur.Une méthode simple et généralement applicable pour la solution de modèles de ce genre est décrite.ZusammenfassungEs wird gezeigt, dass für bestimmte Wärme- und Stoffübertragungsprobleme in der turbulenten Grenzschicht gleichartige Lösungen erzielt werden können, sofern die Wärmeübertragungs- und Stoffübertragungsschichten innerhalb der Schicht konstanter Scherder Fall ist. Geschlossene Lösungen erhält man für den Wärmeübergang von einer Fläche konstanter Temperatur, für den Stoffübergang bei endlicher Grenzflächengeschwindigkeit und für den Wärmeübergang von einer Fläche mit bestimmter Temperaturverteilung. Ein Vergleich experimentell ermittelter Värmeübertragungsdaten bei Rohren mit Spaldingfunktionen, die aufgrund der beiden Spaldinggesetze berechnet wurden, zeigt, dass man mit einem der beiden Gesetze bedeutend bessere Ergebnisse erzielt.
A general method for calculating residence time distributions for systems with internal reflux is described. The method allows the derivation of the Laplace transform of any system composed of mixed vessels with both forward and backward flow between them. In particular, the properties of a linear cascade of mixed vessels with forward and backward flow between the vessels is discussed.RésuméUne méthode générale du calcul des répartitions du temps de résidence pour des systèmes ayant un reflux interne, est décrite. La méthode permet la dérivation de la transformation de Laplace de tout système composé de récipients mixtes ayant entre eux un courant dans les deux sens. En particulier, les propriétés d'une cascade linéaire de récipients mixtes avec courant dans les deux sens entre eux, est discuté.ZusammenfassungEine allgemeine Methode zur Berechnung der Verweilzeitverteilungen für Systeme mit innerem Rückfluss wird beschrieben. Die Methode gestattet die Ableitung der Laplace-Transformierten für jedes System, das sich aus Gefässen mit Vorwärts- und Rückwärtsfluss zwischen ihnen zusammensetzt. Im besonderen werden die Eigenschaften und das Verhalten einer Linearkaskade gemischter Gefässe mit Vorwärts- und Rückwärtsfluss zwischen den Gefässen besprochen.
Hydrodynamic mixing in a porous medium is considered with the help of a model containing a series of perfect mixers with stagnant zones. The relationship between this model and the usual diffusion one is studied. Both models are investigated by the device of injection of a delta-shaped tracer at the entrance to the system. For a large enough number of cells and a sufficiently large length of the porous medium, described by a diffusion model, the tracer concentration distribution at the exit as a function of time is shown to be normal in both cases. The effective coefficient of diffusion, or the dispersion, is found from a comparison of the parameters of these two distributions. At small rates of fluid exchange between the flowing and stagnant zones, the dispersion coefficient is very large. This imposes rigorous limitations on the length of the porous medium necessary for the establishment of a normal concentration distribution at the exit. If, during experimental determination of the dispersion coefficient, these conditions are not fulfilled, i.e. if the porous medium does not prove to be long enough the normal distribution at exit is not established. Then the curve is bell-shaped with a long, steady “tail”. Calculation shows, that the distribution can then be represented as the sum of two distribution: a normal distribution and an exponentially decaying one. The parameters of the porous medium can be determined from experimental results.
Some theoretical equations, regarding the porosity distribution in a fluidized bed of very large diameter, have been developed. Measurements on this porosity distribution in the centre of a fluidized bed of glass beads and air agreed sufficiently well with the theory. In general three distinct zones in the centre of the fluidized bed could be distinguished. Near the sieve, a sieve effect zone was found, followed by a zone of constant porosity reaching up to the initial bed height at incipient fluidization and above it a zone of increasing porosity for which a probility function has been found. The constants in this function have been correlated with some fluidization variables, such as fluidization velocity, bed weight and particle size.
A physical model is proposed for a homogeneous fluidized bed. An equation of motion is established for one of an ensemble of particles in interaction with a fluid. By means of this equation an expression is established for the variance, σε2, of the void fraction, a quantity connected with the mixing process taking place in the fluidizing agent. An expression for the axial diffusion coefficient is established starting from the model suggested here and using the equation derived for σε. This equation is in satisfactory agreement with experimental results obtained by Kramers.
Rates at which solids mix laterally through meshed and unmeshed openings in vertical partitions subdividing a fluidised bed are reported. The effects on mixing of the superficial gas velocity, particle size, and total and fractional free aperture areas are assessed, and bed expansion data reported. The flux rate of material through the apertures was found to be directly proportional to the difference between the superficial and the minimum fluidising velocities, and inversely proportional to a power function of the mean particle diameter. Reducing the free area available for passage of material across the bed decreased the circulation flux rate: increasing the total aperture area had the same effect. These influences on mixing are explained, and the mixing rates and bed expansion data correlated against relevant experimental variables.
The method of moments has been applied to a system of well mixed discrete stages with backflow between adjacent stages. Both differential and difference time dependency have been treated. The recycle rate (back-flow rate) for both cases has been related to the variance of the concentration-time distributions resulting from an arbitrary input of nontransferable tracer.RésuméL'auteur applique la méthode des moments à un système d'étages distincts bien agités avec un reflux entre étages consécutifs.Le problème a été traité en fonction du temps par la méthode des différentielles et des accroissements successifs. La vitesse de recirculation (vitesse de reflux) pour les deux cas, a été reliée à la variation des distributions (concentration-temps) résultant d'une alimentation variable d'un indicateur non enregistreur.ZusammenfassungDie Momentenmethode wurde auf ein System einzelner gut durchmischter Stufen einer Kaskade angewandt, wobei eine Rückvermischung zwischen zwei benachbarten Stufen vorlag. Die Zeitabhängigkeit wurde sowohl durch Differential- als auch durch Differenzengleichungen erfaßt. Die Geschwindigkeit der Rückvermischung wurde in beiden Fällen auf die Varianz der zeitlichen Konzentrationsverteilung, die nach willkürlicher Eingabe eines inerten Tracers entstand, bezogen.
The distribution of residence times in a continuous flow system can be deduced from experiments concerning the behaviour of longitudinal concentration gradients on their course through the system. In this paper the application of sinusoidally varying concentrations is treated from a theoretical and experimental viewpoint. As an illustration of this frequency response analysis, experimental results are given for longitudinal diffusion in liquid flow through packed Raschig rings and for back-mixing of a liquid flowing over the packing of an absorption column.RésuméLa répartition des “durées de séjour” de l'écoulement dans les systèmes en continu peut être déduit de la façon, dont un gradient longitudinal de la concentration se propage à travers le système. Ici on traite l'application des variations sinusoidales de la concentration, du point de vue théorique et experimentale. On présente des résultats expérimentaux de cette analyse harmonique: la diffusion longitudinale dans un liquide qui traverse un lit fixe d'anneaux Raschig et le mélange longitudinal d'un liquide descendant dans une colonne d'absorption à garnissage.
Longitudinal mixing properties were investigated for a water stream flowing through 2 in. and 4 in. tubes packed with 1·3, 3·0 and 3·2 mm spheres over the Reynolds number range 3 to 4500. The method of injecting a step function of salt solution through a simulated plane source was used, coupled with electrical conductivity measurements with very small probes. This method is not limited by the assumption of a flat velocity profile. The statistical model proposed by Einstein is tested and found to be applicable over the entire range of all variables studied. The results are reported in terms of an eddy diffusivity as a function of hydraulic radius and line velocity which are proposed as the characteristic length and velocity terms. The observed eddy diffusivities are somewhat lower than those of other investigators who used radially integrated rather than point measurements of the breakthrough curve. It is shown that the integrated results are too high due to a non-flat velocity profile.
The residence time distribution functions for multistage systems with backmixing are derived. The effect of backmixing on the system behavior is analyzed. In addition, an analytical expression for the probabilities for backmixing is given.
On déduit les fonctions de répartition du temps de séjour moyen dans le cas de systémes multi-étagés où il se fait un mélange de retour; on analyse I'effet de celui-ci sur le système et indique une formule analytique touchant les probabilités dans le cas d'un mélange de retour.
In Wirbelschichten mit kontinuierlichem Feststoffdurchsatz ist die Kenntnis der Feststoffvermischung für reaktionstechnische Belange wichtig. Daher wurden aus Messungen der Verweilzeitverteilungen des durchgesetzten Feststoffes axiale Mischungskoeffizienten ermittelt. Die Wirbelschichten bestanden aus Quarzsand (mittlere Korngröße 250 μm), der mit Luft aufgewirbelt wurde, und hatten einen Durchmesser von 10 cm und Höhen von 8,5; 14,5 und 34,5 cm. Die Mischungskoeffizienten nahmen mit dem Feststoffdurchsatz und der Höhe der Wirbelschicht zu; sie waren jedoch praktisch unabhängig von der Geschwindigkeit des aufwirbelnden Gases. Die Ergebnisse zeigten ferner, daß die Wirbelschicht für die Beschreibung der Feststoffvermischung als ein heterogenes System anzusehen ist.
Settling of small particles in a fluid; mathematical theory.-Small particles immersed in a liquid experience a motion which is the combination of a steady gravitational drift and a Brownian movement. If there are space variations in the density of distribution of particles, the Brownian movement produces a diffusion which tends to equalize the density. In the steady state the density n of particles is an exponential function of x, the distance below the surface of the liquid. This paper investigates the manner in which the steady state is established. A consideration of the combined effect of fall and diffusion leads to a partial differential equation for the number density of particles as a function of depth and time. A set of special solutions is obtained in terms of which a solution satisfying initial and boundary conditions can be expressed. (1) Liquid of finite depth. The solution is obtained for a liquid of finite depth with an arbitrary initial distribution n0=f(x). For the case of uniform initial distribution a reduced form of the solution is obtained which contains a single parameter. This one parameter family of curves is plotted, and from these curves, either directly or by interpolation, may be obtained the density distribution at any time for a solution of any depth, density, and viscosity, and for particles of any size and density. For small values of t, since the solution obtained converges slowly, an image method is used to obtain an integral formula for the density. (2) Liquid of semi-infinite or infinite depth. In the case of a liquid of infinite depth the solution for an arbitrary initial distribution is expressed by the Fourier integral identity. The case of zero initial density for negative x, and constant initial density for positive x is calculated, as is also the case of particles initially uniformly distributed over a layer of depth h. In the case of a liquid extending from x=0 to x=∞, the boundary conditions are satisfied by assuming a suitable fictitious initial distribution over the range from x=-∞ to x=0. The cases of uniform initial distribution, and initial distribution over a layer, are calculated. The latter case, while derived for a liquid of semi-infinite depth, gives approximately the distribution of density during the settling of a layer of particles initially distributed uniformly over a depth h at the upper end of a very long column of liquid.
A method is described to determine quantitatively the change of the heat conductivity in an electric field parallel to the temperature gradient. Measurements were performed on NF3, CHCl3, C2H5Cl, CH3CN and C2H5CN. While the heat conductivity of NF3, CHCl3 and C2H5Cl decreases in an electric field, an increase of the conductivity has been found for CH3CN and C2H5CN.
When a fluid flows through a vessel at a constant rate, either “piston-flow” or perfect mixing is usually assumed. In practice many systems do not conform to either of these assumptions, so that calculations based on them may be inaccurate. It is explained how distribution-functions for residence-times can be defined and measured for actual systems. Open and packed tubes are discussed as systems about which predictions can be made. The use of the distribution-functions is illustrated by showing how they can be used to calculate the efficiencies of reactors and blenders. It is shown how models may be used to predict the distribution of residence-times in large systems.RésuméQuand, dans un récipient, on introduit, à vitesse constante, un fluide donné, on suppose généralement soit un mélange parfait, soit un “écoulement frontal parfait.” En pratique, de nombreux systèmes s'écartent de l'une ou l'autre de ces hypothèses simplificatrices et les calculs qui en résultent sont plus ou moins inexacts. L'auteur expose, pour des systèmes réels, comment l'on peut définir et mesurer des fonctions de distribution pour la “durée de séjour”: ceci peut s'appliquer à des tubes vides ou munis de garnissages. Par emploi de ces fonctions de distribution, l'auteur montre comment on peut calculer l'efficacité des réacteurs ou des mélangeurs. Des modèles peuvent être utilisés pour prévoir la répartition des “durées de séjour” dans des systèmes de grandes dimensions.
Theory is given to predict the value of k in the equation for a bubbling fluidised bed U=kU 0+Q B/A; here U is the superficial fludising velocity, U=U O at incipient fluidisation, Q B is the observed bubble flow, and A the cross-sectional area of the bed. In the simple two-phase theory, k=1; but the theory herein, assuming constant voidage fraction between the bubbles, shows that for a regular array of bubbles, k=1+ε b for a two-dimensional system, provided ε b=volume of all bubbles/bed volume, is small. It is inferred that for a three-dimensional system, k=1+2ε b. Thus the simple two-phase theory, though not strictly correct, is not much in error; the error is likely to be largest when U is not much greater than U O, and particularly just above the distributor of a fluidised bed.
The effect of longitudinal diffusion in chromatographic and ion exchange columns is considered. Calculations made under the assumption of pointwise local equilibrium show that sharp boundaries are smoothed out, thus casting some doubt on the column method for determining isotherms. The problem in which the local rate of removal follows a first order kinetic law is also solved and this solution is a new one.
A catalyst velocity of 11 fps in a pilot unit transfer line was determined by injection of catalyst tagged with zirconium--niobium-95. Similar injections have illustrated mixing patterns in the reactor proper. The examples presented of the types of data which may be obtained demonstrate the unique ability of the tracer technique to provide vital information concerning the effects of operating conditions and structural designs on solids-mixing patterns in fluidized systems. (auth)
Turbulent chemical reactors are modeled by networks of stirred tanks, with the stochastic nature of the mixing introduced by taking the interstage flows to be stationary Markov processes. Some general features of tracer experiments in these quasi-steady flows are discussed, together with their relation to residence time distributions. The statistics of tracer experiments are analyzed, and related on the one hand to the esti-mation of mixing parameters, and on the other hand to the forecast of average yield from the reactor system under first-order kinetics. The variability of the reactor performance and the general story of more complicated kinetic mechanisms are deferred for a later report.
The feasibility of gaseous fluidization of particles in the size range of less than 50 microns was investigated. The ratio of the incipient fluidization velocity, calculated by a conventional relationship without accounting for interparticle forces, to the incipient fluidization velocity, determined by pressure drop and heat transfer measurements, was used as an index, Fl, describing the fluidizability of a particulate material. The results indicate that Fl is closely related to the interparticle adhesive force. The limitations of the feasibility of fluidization depend on the ratio of the weight of a particle to the sum of its weight and adhesive force; no fluidization could be obtained when this ratio was less than 10-3.
Retallick has indicated how to find the residence time distribution for a cascade of n ideal mixers with backmixing ratio α, the ultimate purpose being to derive a from experimental data. For large n when his method is cumbersome, α is obtained more easily from the relative variance, using Van der Laan's formula. This formula is here derived from a set of recurrent relations, constituting a partial difference equation relating moment of distribution to stage number and order of moment. The moments are obtained therefrom as power series in x = α/(1 + α). This treatment is also generalized to cover the case (Bell and Babb) of injection and observation at intermediate points.
A Markoff theory of particle diffusion in homogeneous fluidization is founded on nonlinear Langevin equations and associated quasi-linear and linear stochastic equations, describing the "microscopic" particle-fluid and particle-particle interactions when velocity fluctuations are small and their distribution is Gaussian. Anisotropy is permitted through directional differences in fluctuation energy and particle-fluid friction. Particle and interstitial fluid diffusion are found to be symmetric stochastic processes characterized by a single directional diffusivity sensitive to void fraction and particle-fluid properties. Comparisons of theoretical and experimental diffusivities indicate that considerable anisotropy and inhomogeneity exist during fluidization, attributable to mean velocity distributions and random "macroscopic" disturbances. The stochastic model is then generalized to include fluidized diffusion arising from macroscopic turbulence on the scale of several particle diameters.
Assuming that the inhomogeneities (bubbles) are uniformly distributed inside the fluidized bed, equations are established for the diffusion coefficient of the solid particles and for the heat transfer coefficient between the fluidized bed and the vessel wall.
When there is no backmixing, the probability density for any residence time is given by a single term, but in the case of backmixing, this density is the summation of an infinite series. A particle passing through the cascade can backtrack to the previous vessel any number of times from zero to infinity, and there is a term in the series for each number of backtracks. Described is a computer calculation for generating the coefficients for the terms in the series.
Residence time distribution of solids in single and multicompartment fluidized beds was investigated as a function of solids and gas flow rates. Two types of baffles were investigated - perforated plates and plates equipped with a downcomer. The residence time distribution was represented by the F function and was determined experimentally with the aid of a pulse of magnetic tracer. The residence time distribution for a single stage was represented by F(t) = 1 - e -η (t - ε/θ) for F(t) ≥ 0. The two parameters, η and ε, varied as would be expected from the physical behavior of the system. The results obtained for multistage operation could be calculated by the use of the values obtained for η and ε from single-stage data.
The measurement and analysis of residence time distribution is an important tool in the study of continuous flow systems. A study of available experimental data shows that the usual assumptions of perfect mixing or plug flow do not correspond to the situation existing in real flow systems. The residence time distribution for real systems can be represented by an F-function of the form F(t) = 1 - exp [- η (t - ε/θ)] for t ≥ ε F(t) = 0 for 0 < t ≤ ε This equation results for a number of plausible flow models that include the additional possibilities of dead-space, short-circuiting, error in average residence time determination, and lag in response and any combination of these models. This equation can be used to describe the experimental results obtained for single as well as multistage systems.
A new method for the representation of residence time variability in continuous flow systems is presented. The method is based on the use of the intensity function. The advantage of this method is that it allows a physical insight into the mixing processes within the system and enhances the interpretability of experimental curves. The phenomenon of stagnancy is discussed and defined operationally. Theoretical and practical examples are used to illustrate the usefulness of the concepts introduced.
In this paper a new experimental method, based on the freezing of the fluidized bed by the aid of paraffin wax is used for the determination of the diffusion coefficient of the solid particles. The experimental diffusion coefficients are compared with those given by a theoretical equation. In order to explain the time and spatial dependence of the experimental diffusion coefficient a cinematographic study supplemented by visual observations is also made. One concludes that for small tube diameters a circulation motion of the solid particles takes place which is responsible for the mentioned effect.
Cet article présente une nouvelle méthode expérimentale pour déterminer le coefficient de diffusion de particules solides en figeant un lit fluidisé à l'aide de paraffine. Ces coefficients expérimentaux sont comparés à ceux obtenus théori-quement. Une étude cinématographique, complémentée d'observations visuelles, est utilisée pour expliquer la dépendance dans le temps et l'espace des coefficients de diffusion expérimentaux. Un mouvement circulatoire de la particule solide dans les tubes de faibles diamètres est responsable de l'effet mentionné.
A new method of measuring local solid mixing rates utilizing electrical resistance probe measurements is proposed. Apparatus is described briefly and data obtained from a fluidized conducting-coke bed are given to illustrate its utility.
On propose une nouvelle méthode pour mesurer le taux local de mélange de solids utilisant une électrode à résistance électrique. L'appareil est décrit brièvement et les données obtenucs dans un lit fluidisé de coke sont données pour démontrer son utilité.
Die Tendenz gasdurchströmter Schüttgutschichten, im aufgewirbelten (fluidized) Zustand instabile, turbulente Partikelbewegungszustände zu zeigen — meist als „Inhomogenität” bezeichnet — wurde von verschiedenen Autoren mit der effektiven Zähigkeit des Wirbelschichtbreies in Zusammenhang gebracht. Es gelingt, mit Hilfe der Löchertheorie für Flüssigkeiten eine Formel für diese „Wirbelschichtzähigkeit” zu entwickeln, die die Abhängigkeit von der Partikelgröße, der Schichtexpansion, dem Dichteverhältnis und der Strömungsgeschwindigkeit sinnvoll wiedergibt. Die Übereinstimmung mit groben Messungen ist zufriedenstellend. Aus den energetischen Verhältnissen bei der Bildung von Blasen im Schichtinnern wird ein Inhomogenitätsgrad hergeleitet, der die Wirbelschichtzähigkeit, die Zwischenpartikelreibung und die Schichtexpansion als Faktoren enthält. Die Wirbelschichtzähigkeit läßt sich mit einer von Wilhelm angegebenen Kenngröße und der Inhomogenitätsgrad mit der Froude schen Kennzahl in engen Zusammenhang bringen. Die Zwischenpartikel-Reibungswerte in gas- und flüssigkeitsdurchströmten Wirbelschichten werden durch eine als „Schmierfähigkeit” definierte Größe untereinander verknüpft.
The equilibrium stage concept used in staged contacting operations was adopted as a measure of heat transfer efficiency for a fluidized bed. A simple model was developed which postulates that solids flow through the bed can be by perfect mixing, plug flow and short-circuiting.
Efficiencies were determined experimentally in a 150 mm diameter fluidized bed with air as the fluidizing medium and sand as the solid. Heat transfer efficiencies greater than 100% were obtained indicating that small diameter, low aspect ratio fluidized beds do not behave as perfect mixers. The results indicate that heat transfer measurements can be used to develop information on solids flow behavior.
Le concept de phase d'équilibre qu'on utilise dans les opérations éstagées par contact a été adopté comme mesure du rendement de la transmission de la chaleur dans le cas d'un lit fluidisé. On a mis au point un modèle simple, oú l'on postule que l'écoulement de matières solides dans le lit peut se faire au moyen d'un mélange parfait, d'un écoulement en bloc et d'un court-circuitage.
On détermine les rendements experimentalement dans un lit fluidisé de 150 millimètres de diamètre en employant l'air comme milieu de fluidisation et le sable comme matière solide. On a obtenu des rendements de transmission de la chaleur excédant 100%, ce qui indique que les lits fluidisés de faible diamètre et allongement n'assurent pas un mélange parfait. Les résultats indiquent que les mesures de la transmission de la chaleur peuvent servir à fournir de renseignements sur le comportement de l'écoulement des matières solides.
Motion pictures at 2,000 frames/sec. were used to measure the movements of individual solid particles and gas bubbles in a fluidized bed. Air was used to fluidize 0.028-in. glass spheres and 200-mesh alumina in a 3.75-in. glass column. Aggregates were very common, and each moved as a unit. Particles and aggregates near the wall showed pronounced alternations of fast and slow movements both upward and downward. Individual particles exhibited spin. Baffles increased the bed density and decreased the particle velocities. Small bubbles rose rapidly with little change in shape. Large cavities were slow and tended to collapse and reform elsewhere.
Some characteristics are reported for the fluidization of an air-microspheroidal catalyst system in a 16-in.-diameter bed equipped with baffles. The back-mixing characteristics and retention-time distributions of gas and solids, allowable gas and solids velocities, entrainment rate, and bed density are studied as functions of baffle design.
It is shown that the use of baffles narrows the retention-time spectrum and permits either concurrent or countercurrent flow while not seriously reducing gas or solids throughput or solids holdup.
Models are defined for various mixing conditions, in continuous flow systems. Differential equations are derived which take into account an effective volume of mixing, possible short-circuiting, hold-up time of the system, partial displacement or piston flow.
The values of the different factors contained in the integrated equations can he determined experimentally by the particular response of a given system to a sudden change in composition of the feed.
A correlation of the effective volume of mixing and the agitator r.p.m. is presented.
A design procedure is suggested for tank flow reactors when partial mixing occurs.
Les auteurs définissent des modèles pour différentes conditions d'agitation dans les systèmes à écoulement continu. Ils déduisent des équations différentielles qui tiennent compte d'un volume efficace d'agitation, de court-circuits possibles, du temps de rétention du système et de l'écoulement par déplacement ou “frontal”.
Les valeurs des différents facteurs contenus dans les équations intégrées peuvent ětre obtenues à partir de la réponse particulière d'un système donné à un changement subit de la composition de l'alimentation.
Les auteurs présentent une corrélation entre le volume efficace d'agitation et la vitesse de l'agitateur.
Ils proposent une méthode de calcul pour les réacteurs en régime continu lorsque l'agitation est partielle.
No Abstract. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/37343/1/690120439_ftp.pdf
- van der Laan
Interaction between Fluids and Particles
- Rowe
- Brenner